A theory of spin-orbit coupling in bilayer graphene is presented. The electronic band structure of the AB bilayer in the presence of spin-orbit coupling and a transverse electric field is calculated from first principles using the linearized augmented plane-wave method implemented in the WIEN2K code. The first-principles results around the K points are fitted to a tight-binding model. The main conclusion is that the spin-orbit effects in bilayer graphene derive essentially from the single-layer spin-orbit coupling which comes almost solely from the d orbitals. The intrinsic spin-orbit splitting (anticrossing) around the K points is about 24 mu eV for the low-energy valence and conduction bands, which are closest to the Fermi level, similarly as in the single-layer graphene. An applied transverse electric field breaks space inversion symmetry and leads to an extrinsic (also called Bychkov-Rashba) spin-orbit splitting. This splitting is usually linearly proportional to the electric field. The peculiarity of graphene bilayer is that the low-energy bands remain split by 24 mu eV independently of the applied external field. The electric field, instead, opens a semiconducting band gap separating these low-energy bands. The remaining two high-energy bands are spin split in proportion to the electric field; the proportionality coefficient is given by the second intrinsic spin-orbit coupling, whose value is 20 mu eV. All the band-structure effects and their spin splittings can be explained by our tight-binding model, in which the spin-orbit Hamiltonian is derived from symmetry considerations. The magnitudes of intra- and interlayer couplings-their values are similar to the single-layer graphene ones-are determined by fitting to first-principles results.